There is absolutely no way that you should even think about doing any of this without a complete backup on a separate disk. Verify the backup before beginning.
1) If you have a complete backup, the easiest way to do this would be to wipe everything, make the partitions you want, and then copy the information back from your backup. If your Recovery HD is deleted, there is no way to re-create it without reinstalling the OS. (At least none that I could find when I was searching recently.)
2) Assuming you have a backup OTHER than that Backup partition, if you want to attempt to merge these, I would try it in this order:
a. Delete the 'Backup' partition.
b. Merge 'Data' partition and former 'Backup' partition.
c. Merge System partition and other partition.
d. Re-create the 'Backup' partition and restore it from whatever backup you have.
I have done several live-system partition resizings with no problems. However, the one time that there was a problem, I had to reformat the entire drive and start over.
Try to run Disk utility upon system start up, this may increase functionality. So delete second volume and extend Macintosh HD by dragging it down to make as a single unit. Also repair disk and permissions.
From the screenshot I noticed that you selected your boot disk, which is not possible to edit while OS is running, however second split of HD should be editable. To make less confusion, consider renaming HD, for potential data prevention while modifying HD
Best Answer
To elaborate the answer by @klanomath this is what you must do step-by-step:
get the disk identifier for both disks you want to merge by "diskutil list [disk name]". Choose any of the two disk names, like:
This returns something like
In the IDENTIFIER column, get the disk identifiers for both disks, here: disk3s1 and disk3s2
merge the two disk partitions by "mergePartitions [format] [new disk name][disk identifier 1] [disk identifier 2]". If you want to use the "MacBackup3", you would then enter:
The first disk [disk identifier 1] is the only one that diskutil tries to preserve the data for. If it can't because the disk has no resizable data format, it will give a warning message:
Caution! If you enter "y", your data on [disk identifier 1] will be deleted. This is the last chance to abort by "N". If you have backed up the data, you can go on with "y" - your disk partitions will be merged.